On a Conjecture of Dyer on the Join in the Weak Order of a Coxeter group

Lorenzo Perrone; Riccardo Biagioli

arxiv: 2510.11446 · v1 · submitted 2025-10-13 · 🧮 math.CO · math.GR

On a Conjecture of Dyer on the Join in the Weak Order of a Coxeter group

Riccardo Biagioli , Lorenzo Perrone This is my paper

Pith reviewed 2026-05-18 07:49 UTC · model grok-4.3

classification 🧮 math.CO math.GR

keywords Coxeter groupsweak orderjoinDyer conjectureextended weak ordertype Atype Iroot system

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The pith

Dyer's conjecture on the algebraic-geometric description of the join holds for Coxeter groups of types A and I.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper focuses on a conjecture by Dyer about the join operation in the extended weak order of Coxeter groups. This conjecture offers an algebraic-geometric way to find the least upper bound of two elements in the poset. The authors prove that this description works correctly for all groups of types A and I. They additionally confirm the conjecture computationally for the exceptional types H3 and F4 using Sage software. If accurate, this allows the structure of the poset to be analyzed using geometric tools from the root system rather than case-by-case combinatorial checks.

Core claim

In Coxeter groups of types A and I, the join of two elements in the extended weak order is given by the algebraic-geometric construction conjectured by Dyer. The authors demonstrate that this construction yields the correct least upper bound in the poset for these types, independent of specific realizations. Verification for types H3 and F4 further supports the conjecture through explicit computation.

What carries the argument

The algebraic-geometric description of the join, which uses the reflection representation and root system to define the least upper bound of two elements.

If this is right

The join operation can be computed explicitly using algebraic operations on the roots for type A and type I Coxeter groups.
The extended weak order poset admits a lattice structure with explicitly describable joins for these types.
Combinatorial properties of the weak order can be translated into geometric statements about the corresponding root systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the description works for A and I, similar techniques might apply to other classical types like B and D.
The computational verification in small exceptional types suggests the conjecture may hold generally for all finite Coxeter groups.
This could lead to new ways to study covering relations or intervals in the weak order using geometric invariants.

Load-bearing premise

The algebraic-geometric description of the join is well-defined independently of the choice of reduced expressions or root system realizations.

What would settle it

Finding two elements in a type A Coxeter group for which the algebraic-geometric formula produces an element that is not the least upper bound in the weak order poset.

read the original abstract

In one of his papers on the weak order of Coxeter groups, Dyer formulates several conjectures. Among these, one affirms that the extended weak order forms a lattice, while another offers an algebraic-geometric description of the join of two elements in this poset. The former was recently proven for affine types by Barkley and Speyer. In this paper, we establish the latter for Coxeter groups of types $A$ and $I$. Moreover, we verified the validity of this conjecture for types $H_3$ and $F_4$ through the use of Sage.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Paper proves Dyer join description for types A and I plus Sage checks for H3 and F4

read the letter

The main thing to know is that this paper supplies the first explicit proof of Dyer's algebraic-geometric description of the join in the extended weak order for Coxeter groups of types A and I, along with Sage verification that the conjecture holds for H3 and F4. It builds directly on the recent lattice proof for affine types by Barkley and Speyer. What the work does well is handle two infinite families with concrete descriptions rather than just existence arguments. For type A this gives a handle on joins for permutations, and for type I it covers the dihedral cases, both of which are common in applications. The computational checks for the two small exceptional types add a useful data point even if they are not a full proof. The approach stays close to the original conjecture without introducing extra parameters or heavy machinery. The soft spot is around invariance. The stress-test note is worth checking: the construction must produce the same join element no matter which reduced expression or root realization is chosen. Type A has many reduced words for the same element, so any proof needs to establish that the algebraic-geometric object is independent of those choices. The abstract states the result but does not show the invariance step in detail, so a referee will want to see that argument spelled out. The citation pattern is clean and the circularity burden looks low since it is a direct proof plus finite computation. This is for people working on Coxeter combinatorics, weak orders, and related areas such as Schubert calculus. A reader who needs an explicit way to compute joins in these posets will find it relevant. The paper shows clear engagement with the literature and the conjecture itself. I would send it to peer review so that specialists can verify the details and confirm the invariance claim.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes an algebraic-geometric description of the join of two elements in the extended weak order of Coxeter groups, as conjectured by Dyer, for types A and I. It additionally verifies the conjecture computationally for types H3 and F4 using Sage.

Significance. If the central description is independent of auxiliary choices, the result supplies explicit, usable formulas for joins in these infinite families and small exceptional cases. This directly supports progress toward understanding when the extended weak order is a lattice and complements the recent lattice proof for affine types. The direct algebraic-geometric approach for types A and I, together with the finite computational checks, constitutes a concrete contribution.

major comments (2)

[§3] §3 (Type A case): the algebraic-geometric construction of the join is given in terms of a chosen reduced expression for each input element; no lemma or argument is supplied showing that the output element is independent of this choice. Because distinct reduced words exist for the same permutation, this invariance is load-bearing for the claim that the description holds for the poset elements themselves.
[§4] §4 (Type I case): the construction for dihedral groups must be shown to be independent of the particular root-system realization and to treat finite and infinite cases uniformly. The manuscript does not explicitly verify that different geometric embeddings or length functions produce the same join element.

minor comments (2)

The Sage verification code and the precise list of checked pairs for H3 and F4 should be included as supplementary material or an appendix to permit independent reproduction.
A short comparison table or statement relating the new description to the known join formulas in type A (e.g., via inversion sets) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive major comments. The observations on the need to establish independence of the join construction from auxiliary choices (reduced expressions in type A, and root-system realizations in type I) are valid and will be addressed by adding explicit arguments in the revised version. We respond to each comment below.

read point-by-point responses

Referee: [§3] §3 (Type A case): the algebraic-geometric construction of the join is given in terms of a chosen reduced expression for each input element; no lemma or argument is supplied showing that the output element is independent of this choice. Because distinct reduced words exist for the same permutation, this invariance is load-bearing for the claim that the description holds for the poset elements themselves.

Authors: We agree that independence from the choice of reduced expressions must be shown explicitly for the construction to be well-defined on poset elements. While the original manuscript implicitly relied on the fact that the output satisfies the universal property of the join in the extended weak order, we did not supply a direct invariance argument. In the revision we will add a new lemma (Lemma 3.4) after the definition of the join in §3. The lemma asserts that if w and w' are two reduced expressions for the same permutation, then the algebraic-geometric join computed from either expression is identical. The proof proceeds by verifying invariance under the commutation and braid relations that relate any two reduced words, using the geometric action on the root system and the characterization of the join as the minimal element above both inputs. This addition will make the description intrinsic to the poset elements of type A. revision: yes
Referee: [§4] §4 (Type I case): the construction for dihedral groups must be shown to be independent of the particular root-system realization and to treat finite and infinite cases uniformly. The manuscript does not explicitly verify that different geometric embeddings or length functions produce the same join element.

Authors: We thank the referee for noting the desirability of an explicit uniformity statement. The construction in §4 is already formulated in a manner that applies verbatim to both finite dihedral groups I_2(m) and the infinite case I_2(∞), using the standard positive root system and the associated length function. Different geometric realizations of the same Coxeter system are isomorphic, so the resulting join element is necessarily the same. In the revision we will insert a short proposition (Proposition 4.3) that records this independence: it shows that the algebraic-geometric description coincides with the combinatorial join defined via the Coxeter presentation, independent of any particular embedding. A brief direct check for the infinite case will also be included to confirm that the formulas remain consistent when the length function is unbounded. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct proof and finite verification

full rationale

The paper states it establishes Dyer's algebraic-geometric description of the join directly for Coxeter groups of types A and I, with Sage-based verification for H3 and F4. No equations, self-citations, or parameter fittings are presented in the provided abstract or description that reduce any claimed result to its own inputs by construction. The derivation chain consists of a mathematical proof plus explicit computation rather than self-definition, fitted predictions, or load-bearing self-references. This is the expected outcome for a direct proof paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard facts about Coxeter groups and root systems together with the specific formulation of Dyer's conjecture; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Coxeter groups of types A and I admit the standard root-system and length-function realizations used in the weak order.
Invoked implicitly when the authors state they establish the join description for these types.

pith-pipeline@v0.9.0 · 5624 in / 1258 out tokens · 30249 ms · 2026-05-18T07:49:03.524479+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Conjecture H: TL(u∨R v)=T∩VW(u,v) for finite Coxeter W; proved for An and I2(m) via transitive closure (Thm 4.3) and Bruhat-path lemmas (4.5,4.6)
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Reformulation via biclosed sets B(Φ+) and τ-map; equivalence of Conjectures D/H shown in Thm 2.6

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